Geometria 1014 i

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MATEMÁTICAS BÁSICAS

SEMANA 5

GEOMETRIA

PEB

2014 - I

J. SOTO S.

TEOREMA DE THALES

1

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GEOMETRIA PLANA

1. TEOREMA DE THALES. Tres o ms re!"s #r$e$s% &e"erm'(( so)re &os o ms

re!"s se!("es se*me("os #ro#or!'o($es.

EF

DE

BC

ABT

L L L H

=:

: 321

A

B

C

D

E

+

1 L

2 L

3 L

2

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GEOMETRIA PLANA

TEOREMA DE THALES. COROLARIO 1. To& re!" #r$e$ ,(o &e $os $&os &e ,( "r'(*,$o

&e"erm'( e( $os o"ros &os $&os se*me("os

#ro#or!'o($es.

BA

C

D E

EB

CE

DA

CDT

AB DE H

=:

:

A B

C

D E

CA

EC

CB

DC T

AB DE H

=:

:

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GEOMETRIA PLANA

TEOREMA DE THALES). COROLARIO 2. E( !,$,'er "r'/(*,$o e$ se*me("o ,e ,(e $os #,("os

me&'os &e &os &e s,s $&os es #r$e$o $ "er!er $&o e

'*,$ s, m'"&.

A B

C

M N2

),):

.:

AB MN b AB MN aT

medios puntos son N y M H

=

4

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GEOMETRIA PLANA

APLICACIONES DEL TEOREMA DE THALES1. TEOREMA DE LA BISECTRI INTERIOR. Los $&os ,e &e"erm'(( ,( /(*,$o '("er'or &e ,(

"r'/(*,$o so( #ro#or!'o($es $os se*me("os ,e

&e"erm'( $ )'se!"r' '("er'or &e$ /(*,$o so)re e$ "er!er$&o.

A

B

C

M MC

BM

AC

ABT

Aángulodel erior triz bi AM H

=:

intsec:

5

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GEOMETRIA PLANA

APLICACIONES DEL TEOREMA DE THALES2. TEOREMA DE LA BISECTRI ETERIOR. Los $&os ,e &e"erm'(( ,( /(*,$o e3"er'or &e ,(

"r'(*,$o so( #ro#or!'o($es $os se*me("os ,e

&e"erm'( $ )'se!"r' e3"er'or so)re e$ "er!er $&o.

A

B

C M

CM

AM

BC

ABT

Bángulodel exterior triz bi BM H

=:

sec::

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GEOMETRIA PLANA

SEMEJANA DE TRIANGLOSDE+INICI6N.Dos "r'/(*,$os so( seme7("es s.s.s. "'e(e( s,s "res /(*,$os'*,$es 8 s,s "res $&os 9om:$o*os res#e!"';me("e

#ro#or!'o($es.

<

A

B

C P

=

R

P! ABC

!

BC

P!

AC

P

AB

y

!C B P A

∆≈∆⇔

==

===

,,

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GEOMETRIA PLANA

CRITERIOS DE SEMEJANA

. PRIMER CRITERIO DE SEMEJANA. Dos "r'/(*,$os !,$es,'er so( seme7("es s.s.s. "'e(e(

&os /(*,$os res#e!"';me("e '*,$es.

!oro$r'o. Dos "r'/(*,$os re!"/(*,$os so( seme7("es s.s.s. "'e(e(

,( /(*,$o *,&o '*,$.

>

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GEOMETRIA PLANA

?

E7em#$o. E( !,$,'er "r'/(*,$o% e$ #ro&,!"o &e &os $&oses '*,$ $ #ro&,!"o &e$ &'/me"ro &e $ !'r!,(@ere(!'!'r!,(s!r'"% #or $ $",r re$"'; $ "er!er $&o.

!"a#T

!radiode#$r#uloel enins#rito ABC H

2:

.,:

=∆

α

α

M

A

B

CH

2R

!

)

"

!"a#a

"

!

#

M A#onre#t triang sonambos BMC AHB

DEM%&T

22

)..(

.

=→=→

==∆≈∆ α

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GEOMETRIA PLANA

CRITERIOS DE SEMEJANA

). SEGNDO CRITERIO DE SEMEJANADos "r'/(*,$os so( seme7("es s' "'e(e( ,( /(*,$o '*,$

8 $os $&os ,e $o &e"erm'(( res#e!"';me("e#ro#or!'o($es.

C. TERCER CRITERIO DE SEMEJANA. Dos "r'/(*,$os so( seme7("es s' "'e(e( s,s "res $&os

res#e!"';me("e #ro#or!'o($es.

10

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GEOMETRIA PLANA

RELACIONES METRICAS EN EL TRIANGLO RECTANGLO TEOREMA.S' se "r $ $",r 9% !orres#o(&'e("e $ 9'#o"e(,s e( ,("r'(*,$o re!"/(*,$o e("o(!es se ;er'!( $s s'*,'e("es

#ro#os'!'o(es.

11

)A

B

CH

9

m (

!

222

222

2

2

2

111)

)

)

)

)

)

ba" '

#abe

bnan

a

a

b

bm#

m

#

#

b

d

b"a#"

#

a

b#

mn"n

"

"

mb

BHC AHB ABC a

+=

+=

=→=

=→=

=→=

=→=

∆≈∆≈∆

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GOMETRIA PLANA

AREAS1. AREA DE N PARALELOGRAMO. E$ /re &e ,( #r$e$o*rmo es"/ &&o #or e$ #ro&,!"o &e

$ $o(*'",& &e s, )se #or s, $",r.

COROLARIO AREA DE N TRIANGLO

E$ /re &e ,( "r'(*,$o es '*,$ $ sem'#ro&,!"o &e s,)se #or s, $",r.

12

)

9

)

b" A PA!ALEL%( =

b" A2

1=∆

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GEOMETRIA PLANA

PROPOSICIONES MAS COMNES =E SE ERI+ICAN CONRELACI6N AL ÁREA DE N TRIANGLO.. +orm,$ "r'*o(om"r'! &e$ /re &e ,( "r'(*,$o. E$ /re &e ,( "r'(*,$o es '*,$ $ sem'#ro&,!"o &e &os

$&os !o(!e!,"';os #or e$ se(o &e$ /(*,$o !om#re(&'&oe("re s"os &os $&os

1

A

B

C

9

H)

!

b#senA AT

ABC H

ABC 2

1:

:

=

∆

∆

b#senA A#senA" AHBel En

b" A ABC el En

%N DEM%&T!AC)

ABC

ABC

2

1

,

2

1,

.

=→=∆

=∆

∆

∆

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GEOMETRIA PLANA

). AREA DE N TRIANGLO E=ILATERO

!. OTRAS +ORMLAS PARA EL ÁREA DE N TRIANGLO

14

LL

L

4

32 L A =

∆

A

B

C

R

)

r

!

))()((

2

4

# pb pa p p A

pr A

#ba p&i

!

ab# A

−−−=

=

→++

=

=

∆

∆

∆

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GEOMETRIA PLANA

RELACIONES MAS +RECENTES ENTRE LAS AREAS DE DOS TRIANGLOS.

1.- S' &os "r'/(*,$os "'e(e( s,s $",rs !o(*r,e("es%e("o(!es $ re$!':( e("re s,s /res es '*,$ $ re$!':(e("re s,s )ses.

2.- S' e( ,( "r'(*,$o se "r ,( !e;'(% e("o(!es $re$!':( &e $s /res e( ,e ,e& &';'&'&o e$ "r'(*,$o es

'*,$ $ re$!':( e("re $os se*me("os e( ,e &';'&e $!e;'( $ $&o o#,es"o.

15

2

1

2

121

b

b

A

A""&i =→=

A

B

CM

1 A2 A

MC

AM

A

AT

ABC trianguloel en#e*ianauna BM &ea H

=2

1:

:

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GEOMETRIA PLANA

. S' &os "r'/(*,$os so( seme7("es e("o(!es $ re$!':(e("re s,s /res es '*,$ $ re$!':( e("re $os !,&r&ose("re s,s e$eme("os 9om:$o*os.

1

A

B

C

LH

P

=

RS

9$

M

2

2

2

2

2

1:

,,:

"

H

l

L

A

AT

B P A#on P! ABC &i H

==

==∆≈∆

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GEOMETRIA PLANA

AREAS DE OTRAS REGIONES PLANAS.1. Áre &e$ !,&r&o &e $&o L 2. Áre &e$ rom)o &e &'*o($es & 8 D. Áre &e$ "r#e!'o &e )ses ) 8 B% 8 $",r 9

4. Áres &e$ !'r!,$o &e r&'o R.5. Áre &e$ "r#e!'o !'r!,$r. Áre &e$ se*me("o !'r!,$r

1<

2 L A =

Dxd A !%MB%2

1=

"b B

AT!APEC)%

+=

2

( ) ( )r ! L L Ar ! A #ir#trap#ir#trap −

+=−=

2,

360

21

..

22

..

πθ

2. ! AC)!C π =

BC

AD

R

r

A%B A%BC)!C DEL&ECT A!EA AMBC)!C &E( A A A ∆−= ..:..

A

B

MO

ABCDC)!C+LA!T!APEC)% : AMBC)!C+LA!&E(MENT% :

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GEOMETRIA DEL ESPACIO

EL PRISMADE+INICION.Se &e(om'( s,#er!'e #r'sm/"'! $ s,#er!'e *e(er&#or ,( re!" &e(om'(& re!" *e(er"r' ,e se m,e;e

#r$e$me("e s' m'sm 8 se &es#$ so)re ,( #o$'*o($&e(om'(& &'re!"r'.

1>

dire#triz

generatriz

Bbase :

"altura

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DE+INICION.Se &e(om'(( #r'sms re!"os !,(&o $s r's"s $"er$es so(#er#e(&'!,$res $s )ses% 8 !,(&o (o o!,rre s se&e(om'(( #r'sms o)$'!,os.

DE+INICION.E$ #r'sm se &e(om'( #r'sm re*,$r !,(&o e$ #o$*o(o &e$ )se es ,( #o$*o(o re*,$r% e( !so !o("rr'o se &e(om'(#r'sm 'rre*,$r.DE+INICION.

E$ (om)re &e$ #r'sm es &e !,er&o $ (om)re &e$ #o$*o(o &e$ )se% s se &e(om'(r/ #r'sm "r'(*,$r% !,&r(*,$r%#e("*o($ e3*o($ %e"!.

1?

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AREA LATERAL TOTAL F OLMEN DE N PRISMAE$ /re $"er$ &e ,( #r'sm o)$'!,o es '*,$ $ #erme"ro &e $se!!':( re!" #or $ r's" $"er$.E$ /re $"er$ &e ,( #r'sm re!"o es '*,$ $ #ro&,!"o &e s,

$",r #or e$ #erme"ro &e $ )se.EL ;o$,me( &e ,( #r'sm re!"o o)$'!,o es '*,$ $ #ro&,!"o&e$ /re &e $ )se #or s, $",r.DE+INICION.Se &e(om'( !,)o o e3e&ro $ #r'sm !,8s )ses 8 !rs

$"er$es so( !,&r&os &e $&o .

20

3

2

)(

2

)(

6

4

a,

a A

a A

#ubo

#uboT

#ubol

=

=

=

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LA PIRAMIDE.DE+INICION.Se &e(om'( s,#er!'e #'rm'&$ $ s,#er!'e *e(er& #or,( se*me("o &e re!"% ,(o &e !,8os e3"remos s'em#re #s

#or ,( #,("o 7o 8 e$ o"ro e3"remo se &es#$ so)re ,( #o$'*o($.

21

piramideladearista

piramideladeapotema

.. piraladealt

baseladeapot .

A

B C

D

E

MH

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DE+INICIONES.E$ (om)re &e ,( #'r/m'&e es &e !,er&o s, )se.P'r/m'&e re*,$r % es ,e$$ ,e "'e(e #or )se ,( #o$*o(ore*,$r % 8 e$ #'e &e s, $",r !o'(!'&e !o( e$ !e("ro &e $ )se.

Tro(!o &e #'r/m'&e% o #'r/m'&e "r,(!& es $ #r"e &e $#'r/m'&e !om#re(&'& e("re $ )se 8 ,( #$(o ,e !or" "o&s s,s r's"s $"er$es.C,(&o $ se!!':( es #r$e$ $ )se% res,$" ,( "ro(!o &e#'r/m'&e &e )ses #r$e$s. S, $",r es $ &'s"(!' e("re $s)ses.

Tro(!o &e #'r/m'&e re*,$r % es $ #or!':( &e #'r/m'&e re*,$r!om#re(&'& e("re $ )se 8 ,( se!!':( #r$e$ $ m'sm.Ls !rs $"er$es so( "r#e!'os 's:s!e$es '*,$es% 8 $ $",r &e!& ,(o &e e$$os se &e(om'( #o"em &e$ "ro(!o.

22

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AREAS F OLMEN DE NA PIRAMIDE.E$ /re $"er$ &e ,( #'r/m'&e re*,$r% es '*,$ $ sem'#ro&,!"o&e$ #erme"ro &e $ )se #or e$ #o"em &e $ #'r/m'&e.E$ /re $"er$ &e$ "ro(!o &e #'r/m'&e re*,$r es '*,$ $sem's,m &e $os #erme"ros &e $s )ses #or $ me&'& &e$#o"em &e$ "ro(!o.o$,me( &e $ #'r/m'&e es '*,$ ,( "er!'o &e$ /re &e $ )se#or s, $",r.

o$,me( &e$ "ro(!o &e #'r/m'&e s' 9 es $ $",r &e$ "ro(!o 8 )%8 B so( $s /res &e s,s )ses e("o(!es e$ ;o$,me( es" &&o#or

2

B", P)!AM 3

1. =

( )bB Bb", DEP)!AM T!%NC% ++=3

.

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EL TETRAEDRO REGLAR.Se &e(om'( "e"re&ro re*,$r $ #'r/m'&e re*,$r !,8)se 8 !rs $"er$es so( "r'/(*,$os e,'$/"eros.

24

A M

N

H

9

3

2,2

12

3

433

3

2

2

a"a

,

a A

a A

T

l

==

=

=

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1. AREAS F OLMEN DEL CILINDRO CIRCLAR RECTO.

2. AREAS F OLMEN DEL CONO CIRCLAR RECTO

25

r

9 "r ,

r r" A

r" A

T

L

2

22

2

π

π π

π

=

+=

=

9

r

*

"r ,

r rg A

rg A

T

l

2

2

3

1π

π π

π

=

+=

=

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SPER+ICIE ES+ERICA

2

R

3

.

2

..

3

4

4

!,

! A

E&F

E&F &+P

π

π

=

=

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Geometria 1014 i